Is this positive operator Hermitian?

Question

Let $\Omega_1$ and $\Omega_2$ be topological spaces. Let $P: C(\Omega_1)\longrightarrow C(\Omega_2)$ be a Positive linear map. Does that imply $P$ is Hermitian? i.e does $\overline {P(f)}=P(\bar{f})$ hold for every $f\in C(\Omega_1)$?

Answer 1

Yes. Every $f\in C(\Omega_1)$ can be written as $$f=f_1-f_2+i(f_3-f_4)$$ with $f_j\geq0$, $j=1,\ldots,4$. Then $$ P(\bar f)=P(f_1-f_2-i(f_3-f_4))=Pf_1-Pf_2-i(Pf_3-Pf_4)=\overline{P(f)}, $$ since $Pf_1-Pf_2$ and $Pf_3-Pf_4$ are real.